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arXiv · 2412.05217

Stochastic homogenisation of nonlinear minimum-cost flow problems

Peter Gladbach·Jan Maas·Lorenzo Portinale
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Abstract

This paper deals with the large-scale behaviour of nonlinear minimum-cost flow problems on random graphs. In such problems, a random nonlinear cost functional is minimised among all flows (discrete vector-fields) with a prescribed net flux through each vertex. On a stationary random graph embedded in $\mathbb{R}^d$, our main result asserts that these problems converge, in the large-scale limit, to a continuous minimisation problem where an effective cost functional is minimised among all vector fields with prescribed divergence. Our main result is formulated using $Γ$-convergence and applies to multi-species problems. The proof employs the blow-up technique by Fonseca and Müller in a discrete setting. One of the main challenges overcome is the construction of the homogenised energy density on random graphs without a periodic structure.

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Peter Gladbach, Jan Maas, Lorenzo Portinale. 2025-06-26. Stochastic homogenisation of nonlinear minimum-cost flow problems. https://arxiv.org/abs/2412.05217

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